PSI - Issue 8

J. Srnec Novak et al. / Procedia Structural Integrity 8 (2018) 174–183 Author name / Structural Integrity Procedia 00 (2017) 000–000

176

3

2. Cyclic plasticity models

Combined material model (nonlinear kinematic + nonlinear isotropic) is able to capture monotonic elasto-plastic and cyclic hardening/softening behavior of a material. The von Mises yield surface is expressed in Chaboche (2008) as:

( 2 3 0 = − − − − = σ R f σ α σ α ) ( ' ' ' : ' )

0

(1)

where σ ´ and α ´ are the deviatoric stress tensor and the back stress tensor, respectively, R is the drag stress and σ 0 is the initial yield stress. Kinematic part is controlled by α (translation of the yield surface), while isotropic part is related to R , which controls the homothetic expansion of the yield surface during cyclic loading. According to Chaboche (2008), the increment of the back stress, d α , is expressed as a function of the increment of plastic strain, d ε pl , and accumulated plastic strain, d ε pl,acc :

α α = ∑

3 , d 2 α i =

d C ε

γ i i α

(2)

d

−

ε

i

i

pl

pl,acc

i

where C is the initial hardening modulus; the recall parameter γ controls decrease rate of the initial hardening modulus as the plastic strain accumulates. Chaboche model is a superposition of several nonlinear kinematic models. The model with one pair ( C 1 , γ 1 ) is known as the Armstrong and Frederick model. Furthermore, considering γ =0 the Prager model (i.e. linear kinematic) is obtained and relation (2) can be expressed as:

(3)

3 d 2 α

ε

C =

lin d

pl

Expansion of the yield surface is controlled by the nonlinear isotropic model: ( ) pl,acc d d ε R b R R = − ∞

(4)

where b defines the speed of stabilization and R ∞ is the saturation value of the yield surface. R ∞ can be either positive or negative, giving rise to cyclic hardening or softening behavior, respectively. Integration of (4) gives a relationship between R and ε pl,acc : ( ) pl, acc 1 ε b R R e − ∞ = − (5) A stabilized condition is obtained when R reaches R ∞ . Hardening/softening kinetics is mainly governed by the speed of stabilization b and the accumulated plastic strain ε pl,acc . If the amount of the accumulated plastic strain is relatively small, a huge number of cycles in needed to obtain R = R ∞ . Since the accumulated plastic strain depends on loading conditions and cannot be changed, the only available parameter that can be modified to accelerate material stabilization is the speed of stabilization b . Models with increased b parameter, known as “accelerated models”, were firstly introduced in Chaboche and Cailletaud (1986). It was proposed to use a speed of stabilization in the range of (50÷150) b .